Character table for the D28h point group

D28h     E        2 C28    2 C14    2 C28^3  2 C7     2 C28^5  2 C14^3  2 C4     2 C7^2   2 C28^9  2 C14^5  2 C28^11 2 C7^3   2 C28^13 C2       14 C2'   14 C2"   i        2 S28    2 S14    2 S28^3  2 S7     2 S28^5  2 S14^3  2 S4     2 S7^2   2 S28^9  2 S14^5  2 S28^11 2 S7^3   2 S28^13 sh       14 sv    14 sd      <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1g     1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000     ... ... ....T ....... ........T ........... ............T
A2g     1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000 -1.00000 -1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000 -1.00000 -1.00000     ..T ... ..... ....... ......... ........... .............
B1g     1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000  1.00000 -1.00000     ... ... ..... ....... ......... ........... .............
B2g     1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000     ... ... ..... ....... ......... ........... .............
E1g     2.00000  1.94986  1.80194  1.56366  1.24698  0.86777  0.44504  0.00000 -0.44504 -0.86777 -1.24698 -1.56366 -1.80194 -1.94986 -2.00000  0.00000  0.00000  2.00000 -1.94986 -1.80194 -1.56366 -1.24698 -0.86777 -0.44504  0.00000  0.44504  0.86777  1.24698  1.56366  1.80194  1.94986 -2.00000  0.00000  0.00000     TT. ... ..TT. ....... ......TT. ........... ..........TT.
E2g     2.00000  1.80194  1.24698  0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504  0.44504  1.24698  1.80194  2.00000  0.00000  0.00000  2.00000  1.80194  1.24698  0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504  0.44504  1.24698  1.80194  2.00000  0.00000  0.00000     ... ... TT... ....... ....TT... ........... ........TT...
E3g     2.00000  1.56366  0.44504 -0.86777 -1.80194 -1.94986 -1.24698  0.00000  1.24698  1.94986  1.80194  0.86777 -0.44504 -1.56366 -2.00000  0.00000  0.00000  2.00000 -1.56366 -0.44504  0.86777  1.80194  1.94986  1.24698  0.00000 -1.24698 -1.94986 -1.80194 -0.86777  0.44504  1.56366 -2.00000  0.00000  0.00000     ... ... ..... ....... ..TT..... ........... ......TT.....
E4g     2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  0.00000  0.00000  2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  0.00000  0.00000     ... ... ..... ....... TT....... ........... ....TT.......
E5g     2.00000  0.86777 -1.24698 -1.94986 -0.44504  1.56366  1.80194  0.00000 -1.80194 -1.56366  0.44504  1.94986  1.24698 -0.86777 -2.00000  0.00000  0.00000  2.00000 -0.86777  1.24698  1.94986  0.44504 -1.56366 -1.80194  0.00000  1.80194  1.56366 -0.44504 -1.94986 -1.24698  0.86777 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... ..TT.........
E6g     2.00000  0.44504 -1.80194 -1.24698  1.24698  1.80194 -0.44504 -2.00000 -0.44504  1.80194  1.24698 -1.24698 -1.80194  0.44504  2.00000  0.00000  0.00000  2.00000  0.44504 -1.80194 -1.24698  1.24698  1.80194 -0.44504 -2.00000 -0.44504  1.80194  1.24698 -1.24698 -1.80194  0.44504  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... TT...........
E7g     2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  0.00000  2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E8g     2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000  0.00000  0.00000  2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E9g     2.00000 -0.86777 -1.24698  1.94986 -0.44504 -1.56366  1.80194  0.00000 -1.80194  1.56366  0.44504 -1.94986  1.24698  0.86777 -2.00000  0.00000  0.00000  2.00000  0.86777  1.24698 -1.94986  0.44504  1.56366 -1.80194  0.00000  1.80194 -1.56366 -0.44504  1.94986 -1.24698 -0.86777 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E10g    2.00000 -1.24698 -0.44504  1.80194 -1.80194  0.44504  1.24698 -2.00000  1.24698  0.44504 -1.80194  1.80194 -0.44504 -1.24698  2.00000  0.00000  0.00000  2.00000 -1.24698 -0.44504  1.80194 -1.80194  0.44504  1.24698 -2.00000  1.24698  0.44504 -1.80194  1.80194 -0.44504 -1.24698  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E11g    2.00000 -1.56366  0.44504  0.86777 -1.80194  1.94986 -1.24698  0.00000  1.24698 -1.94986  1.80194 -0.86777 -0.44504  1.56366 -2.00000  0.00000  0.00000  2.00000  1.56366 -0.44504 -0.86777  1.80194 -1.94986  1.24698  0.00000 -1.24698  1.94986 -1.80194  0.86777  0.44504 -1.56366 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E12g    2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000  0.00000  0.00000  2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E13g    2.00000 -1.94986  1.80194 -1.56366  1.24698 -0.86777  0.44504  0.00000 -0.44504  0.86777 -1.24698  1.56366 -1.80194  1.94986 -2.00000  0.00000  0.00000  2.00000  1.94986 -1.80194  1.56366 -1.24698  0.86777 -0.44504  0.00000  0.44504 -0.86777  1.24698 -1.56366  1.80194 -1.94986 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
A1u     1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000     ... ... ..... ....... ......... ........... .............
A2u     1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000  1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000  1.00000  1.00000     ... ..T ..... ......T ......... ..........T .............
B1u     1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000  1.00000 -1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000 -1.00000  1.00000     ... ... ..... ....... ......... ........... .............
B2u     1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000  1.00000 -1.00000     ... ... ..... ....... ......... ........... .............
E1u     2.00000  1.94986  1.80194  1.56366  1.24698  0.86777  0.44504  0.00000 -0.44504 -0.86777 -1.24698 -1.56366 -1.80194 -1.94986 -2.00000  0.00000  0.00000 -2.00000  1.94986  1.80194  1.56366  1.24698  0.86777  0.44504  0.00000 -0.44504 -0.86777 -1.24698 -1.56366 -1.80194 -1.94986  2.00000  0.00000  0.00000     ... TT. ..... ....TT. ......... ........TT. .............
E2u     2.00000  1.80194  1.24698  0.44504 -0.44504 -1.24698 -1.80194 -2.00000 -1.80194 -1.24698 -0.44504  0.44504  1.24698  1.80194  2.00000  0.00000  0.00000 -2.00000 -1.80194 -1.24698 -0.44504  0.44504  1.24698  1.80194  2.00000  1.80194  1.24698  0.44504 -0.44504 -1.24698 -1.80194 -2.00000  0.00000  0.00000     ... ... ..... ..TT... ......... ......TT... .............
E3u     2.00000  1.56366  0.44504 -0.86777 -1.80194 -1.94986 -1.24698  0.00000  1.24698  1.94986  1.80194  0.86777 -0.44504 -1.56366 -2.00000  0.00000  0.00000 -2.00000  1.56366  0.44504 -0.86777 -1.80194 -1.94986 -1.24698  0.00000  1.24698  1.94986  1.80194  0.86777 -0.44504 -1.56366  2.00000  0.00000  0.00000     ... ... ..... TT..... ......... ....TT..... .............
E4u     2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  0.00000  0.00000 -2.00000 -1.24698  0.44504  1.80194  1.80194  0.44504 -1.24698 -2.00000 -1.24698  0.44504  1.80194  1.80194  0.44504 -1.24698 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ..TT....... .............
E5u     2.00000  0.86777 -1.24698 -1.94986 -0.44504  1.56366  1.80194  0.00000 -1.80194 -1.56366  0.44504  1.94986  1.24698 -0.86777 -2.00000  0.00000  0.00000 -2.00000  0.86777 -1.24698 -1.94986 -0.44504  1.56366  1.80194  0.00000 -1.80194 -1.56366  0.44504  1.94986  1.24698 -0.86777  2.00000  0.00000  0.00000     ... ... ..... ....... ......... TT......... .............
E6u     2.00000  0.44504 -1.80194 -1.24698  1.24698  1.80194 -0.44504 -2.00000 -0.44504  1.80194  1.24698 -1.24698 -1.80194  0.44504  2.00000  0.00000  0.00000 -2.00000 -0.44504  1.80194  1.24698 -1.24698 -1.80194  0.44504  2.00000  0.44504 -1.80194 -1.24698  1.24698  1.80194 -0.44504 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E7u     2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  0.00000 -2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000 -2.00000  0.00000  2.00000  0.00000  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E8u     2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000  0.00000  0.00000 -2.00000  0.44504  1.80194 -1.24698 -1.24698  1.80194  0.44504 -2.00000  0.44504  1.80194 -1.24698 -1.24698  1.80194  0.44504 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E9u     2.00000 -0.86777 -1.24698  1.94986 -0.44504 -1.56366  1.80194  0.00000 -1.80194  1.56366  0.44504 -1.94986  1.24698  0.86777 -2.00000  0.00000  0.00000 -2.00000 -0.86777 -1.24698  1.94986 -0.44504 -1.56366  1.80194  0.00000 -1.80194  1.56366  0.44504 -1.94986  1.24698  0.86777  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E10u    2.00000 -1.24698 -0.44504  1.80194 -1.80194  0.44504  1.24698 -2.00000  1.24698  0.44504 -1.80194  1.80194 -0.44504 -1.24698  2.00000  0.00000  0.00000 -2.00000  1.24698  0.44504 -1.80194  1.80194 -0.44504 -1.24698  2.00000 -1.24698 -0.44504  1.80194 -1.80194  0.44504  1.24698 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E11u    2.00000 -1.56366  0.44504  0.86777 -1.80194  1.94986 -1.24698  0.00000  1.24698 -1.94986  1.80194 -0.86777 -0.44504  1.56366 -2.00000  0.00000  0.00000 -2.00000 -1.56366  0.44504  0.86777 -1.80194  1.94986 -1.24698  0.00000  1.24698 -1.94986  1.80194 -0.86777 -0.44504  1.56366  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E12u    2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000  0.00000  0.00000 -2.00000  1.80194 -1.24698  0.44504  0.44504 -1.24698  1.80194 -2.00000  1.80194 -1.24698  0.44504  0.44504 -1.24698  1.80194 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E13u    2.00000 -1.94986  1.80194 -1.56366  1.24698 -0.86777  0.44504  0.00000 -0.44504  0.86777 -1.24698  1.56366 -1.80194  1.94986 -2.00000  0.00000  0.00000 -2.00000 -1.94986  1.80194 -1.56366  1.24698 -0.86777  0.44504  0.00000 -0.44504  0.86777 -1.24698  1.56366 -1.80194  1.94986  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.949855824364 = 2*cos(2*π/28) = 2*cos(π/14)
                               1.801937735805 = 2*cos(4*π/28) = 2*cos(π/7)
                               1.563662964936 = 2*cos(6*π/28) = 2*cos(3*π/14)
                               1.246979603717 = 2*cos(8*π/28) = 2*cos(2*π/7)
                               0.867767478235 = 2*cos(10*π/28) = 2*cos(5*π/14)
                               0.445041867913 = 2*cos(12*π/28) = 2*cos(3*π/7)



 Symmetry of Rotations and Cartesian products

A1g  d+g+i+k+m    z2, z4, z6 
A2g  R            Rz 
E1g  R+d+g+i+k+m  {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} 
E2g  d+g+i+k+m    {x2y2, xy}, {z2(x2y2), xyz2}, {z4(x2y2), xyz4} 
E3g  g+i+k+m      {xz(x2−3y2), yz(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4g  g+i+k+m      {(x2y2)2−4x2y2, xy(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E5g  i+k+m        {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E6g  i+k+m        {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 
E7g  k+m 
E8g  k+m 
E9g  m 
E10g m 
A2u  p+f+h+j+l    z, z3, z5 
E1u  p+f+h+j+l    {x, y}, {xz2, yz2}, {xz4, yz4} 
E2u  f+h+j+l      {z(x2y2), xyz}, {z3(x2y2), xyz3} 
E3u  f+h+j+l      {x(x2−3y2), y(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)} 
E4u  h+j+l        {z((x2y2)2−4x2y2), xyz(x2y2)} 
E5u  h+j+l        {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)} 
E6u  j+l 
E7u  j+l 
E8u  l 
E9u  l 

 Notes:

    α  The order of the D28h point group is 112, and the order of the principal axis (C28) is 28. The group has 34 irreducible representations.

    β  The D28h point group is generated by three symmetry elements that are canonically chosen C28, C2 and i.
       Other choices include σh instead of i, or any of C2, σv or σd instead of C2. Also, some ternary combinations of C2, C2, σv and σd act as generators.
       Lastly, the S28 can be chosen, together with i or σh and any one of C2, C2, σv or σd.

    γ  There are two different sets of twofold symmetry axes perpendicular to the principal axis (z axis in standard orientation).
       By convention, the set denoted as C2 contains both the x and y axes.

    δ  There are two different sets of symmetry planes containing the principal axis (z axis in standard orientation).
       By convention, the set denoted as σv contains both the xz and the yz planes.

    ε  The lowest nonvanishing multipole moment in D28h is 4 (quadrupole moment).

    ζ  This point group is non-Abelian (some symmetry operations are not commutative).
       Therefore, the character table contains multi-membered classes and degenerate irreducible representations.

    η  Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6.
       These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1.
       All characters are algebraic integers of a degree much less than half the order of the principal axis.

    θ  The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible
       if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given
       by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three.

    ι  The regular heptagon is the lowest regular polygon that cannot be constructed by compass and ruler alone, which has mystified
       mathematicians since antiquity. The reason for the inconstructibility of the heptagon is that cos(2π/7) has an algebraic degree of 3 and
       thus can not be represented by square roots and integer numbers. An algebraic representation becomes possible if cubic roots and complex numbers
       are used: 2*cos(2π/7) = (328+i*84*√3 + 328−i*84*√3 − 2)/6. This complex expression for a real value is hardly useful.
       Therefore, regular polygons of order 14,21,28,35,42 etc. are also inconstructible, and their cosines have no representation in real radicals.

This Character Table for the D28h point group was created by Gernot Katzer.

For other groups and some explanations, see the Main Page.